# 18.12 Set Functions for Finite Fields

Finite fields are of course domains. Thus all set theoretic functions are applicable to finite fields (see chapter Domains). This section gives further comments on the definitions and implementations of those functions for finite fields. All set theoretic functions not mentioned here are not treated specially for finite fields.

`Elements`

The elements of a finite field are computed using the fact that the finite field is a vector space over its prime field.

`in`

The membership test is of course very simple, we just have to test whether the element is a finite field element with `IsFFE`, whether it has the correct characteristic with `CharFFE`, and whether its degree divides the degree of the finite field with `DegreeFFE` (see IsFFE, CharFFE, and DegreeFFE).

`Random`

A random element of GF(p^n) is computed by computing a random integer i from [0..p^n-1] and returning 0*Z(p) if i = 0 and Z(p^n)^{i-1} otherwise.

`Intersection`

The intersection of GF(p^n) and GF(p^m) is the finite field GF(p^{Gcd(n,m)}), and is returned as finite field record.

GAP 3.4.4
April 1997